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The equation is:

\[
x^3 + (m + 1)x^2 + 2mx + m + 3 = 0
\]

with \( x_1, x_2, x_3 \) being the roots of this cubic equation. We are asked to calculate:

\[
(x_1 + 1)(x_2 + 1)(x_3 + 1)
\]

### Step-by-Step Approach

#### 1. Using Vieta's Formulas

Vieta's formulas for the roots of a cubic equation \( x^3 + ax^2 + bx + c = 0 \) give the following relationships:

- \( x_1 + x_2 + x_3 = -(m + 1) \) (coefficient of \( x^2 \) with opposite sign)
- \( x_1x_2 + x_2x_3 + x_3x_1 = 2m \) (coefficient of \( x \))
- \( x_1x_2x_3 = -(m + 3) \) (constant term with opposite sign)

#### 2. Expression for \((x_1 + 1)(x_2 + 1)(x_3 + 1)\)

We can expand the product \((x_1 + 1)(x_2 + 1)(x_3 + 1)\):

\[
(x_1 + 1)(x_2 + 1)(x_3 + 1) = x_1x_2x_3 + (x_1x_2 + x_2x_3 + x_3x_1) + (x_1 + x_2 + x_3) + 1
\]

Substitute the values from Vieta's formulas:

\[
= -(m + 3) + 2m + -(m + 1) + 1
\]

Simplify:

\[
= -m - 3 + 2m - m - 1 + 1
\]
\[
= -m - 3 + 2m - m - 1 + 1 = -3
\]

Thus, the value of \((x_1 + 1)(x_2 + 1)(x_3 + 1)\) is \(-3\).

### Answer: 

\[
(x_1 + 1)(x_2 + 1)(x_3 + 1) = -3
\]

Would you like more details on this solution or the next problem?

Here are five questions related to this problem:

1. What are Vieta's formulas, and how are they derived for polynomial equations?
2. How does the relationship between the roots and coefficients of a cubic equation help solve problems like this?
3. Can the expression \((x_1 + 1)(x_2 + 1)(x_3 + 1)\) be generalized for higher-degree polynomials?
4. What is the significance of the constant term in a cubic equation when determining the product of the roots?
5. How would the calculation change if the equation had complex roots?

**Tip:** When solving polynomial equations, using Vieta's relations can simplify complex expressions involving sums and products of roots.

Si x1, x2, x3 son las raíces de la ecuación: x^3 + (m + 1)x^2 + 2mx + m + 3 = 0. Calcula: (x1 + 1)(x2 + 1)(x3 + 1).

Learn how to calculate (x1 + 1)(x2 + 1)(x3 + 1) for a cubic equation x^3 + (m + 1)x^2 + 2mx + m + 3 = 0 using Vieta's formulas. This problem involves algebraic concepts like root relationships and polynomial expansions, suitable for Grades 10-12.

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